STATISTICS: MODULE Chapter 3  Bivariate or joint probability distributions


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1 STATISTICS: MODULE Chapte  Bivaiate o joit pobabilit distibutios I this chapte we coside the distibutio of two adom vaiables whee both adom vaiables ae discete (cosideed fist) ad pobabl moe impotatl whee both adom vaiables ae cotiuous. Bivaiate o joit distibutios model the wa two adom vaiables va togethe. A. DISCRETE VARIABLES Eample. Hee we have a pobabilit model of the demad ad suppl of a peishable commodit. The pobabilit model/distibutio is defied as follows: Suppl of commodit (SP) Demad fo commodit (D) This is kow as a discete bivaiate o joit pobabilit distibutio sice thee ae two adom vaiables which ae "demad fo commodit (D)" ad "suppl of commodit (SP)". The sample space S cosists of 5 outcomes (d, s) whee d ad s ae the values of D ad SP. The pobabilities i the table ae joit pobabilities, amel P( D d ad SP s) o P( D d SP s) usig set otatio. Eamples Note: The sum of the 5 pobabilities is.. Joit pobabilit fuctio Suppose the adom vaiables ae ad, the the joit pobabilit fuctio is deoted b p ad is defied as follows: p P( ad ) o P( )
2 ,. Also p( ). Magial pobabilit distibutios The magial distibutios ae the distibutios of ad cosideed sepaatel ad model how ad va sepaatel fom each othe. Suppose the pobabilit fuctios of ad ae p ( ) ad p ( ) espectivel so that p ( ) P( ) ad p ( ) Also p ( ) ad p ( ). P( ) It is quite staightfowad to obtai these these fom the joit pobabilit distibutio p p, p p, sice ( ) ( ) ad ( ) ( ) I egessio poblems we ae ve iteested i coditioal pobabilit distibutios such as the coditioal distibutio of give ad the coditioal distibutio of give.4 Coditioal pobabilit distibutios The coditioal pobabilit fuctio of give is deoted b p( ) is defied as p( ) P( ) ( ) P( ) P ad ( ) p wheeas the coditioal pobabilit fuctio of give is deoted b p( ) ad defied as p( ) P( ) ( ) P( ) P ad p ( ) p p.5 Joit pobabilit distibutio fuctio The joit (cumulative) pobabilit distibutio fuctio (c.d.f.) is deoted b F(, ) ad is defied as F(, ) P( ad ) ad 0 F(, ) The magial c.d.f s ae deoted b F ( ) ad F ( ) F ( ) P( ) ad F ( ) (see Chapte, sectio. ). P( ) ad ae defied as follows
3 .6 Ae ad idepedet? If eithe (a) F(, ) F ( ). F ( ) o (b)p(, ) p ( ). p ( ) the ad ae idepedet adom vaiables. Eample. The joit distibutio of ad is (a) Fid the magial distibutios of ad. (b) Fid the coditioal distibutio of give 0. (c) Ae ad idepedet? B. CONTINUOUS VARIABLES.7 Joit pobabilit desit fuctio The joit p.d.f. is deoted b f (, ) (whee f (, ) 0 all ad ) ad defies a pobabilit suface i dimesios. Pobabilit is a volume ude this suface ad the total volume ude the p.d.f. suface is as the total pobabilit is i.e. f d d d b ad P( a b ad c d ) c a f d d As befoe with discete vaiables, the magial distibutios ae the distibutios of ad cosideed sepaatel ad model how ad va sepaatel fom each othe. Wheeas with discete adom vaiables we speak of magial pobabilit fuctios, with cotiuous adom vaiables we speak of magial pobabilit desit fuctios. Eample. A electoics sstem has oe of each of two diffeet tpes of compoets i joit opeatio. Let ad deote the adom legths of life of the compoets of tpe ad, espectivel. Thei joit desit fuctio is give b ( ) / f (, ) e ; + > 0 > othewise
4 4 Eample.4 The adom vaiables ad have a bivaiate omal distibutio if ( ) f, ae b whee a π σ σ ρ ad b ( ρ ) µ µ µ µ ρ + σ σ σ σ whee < <, < <, < µ <, < µ <, < ρ <, σ > 0, σ > 0. The p.d.f. suface is show below ad as ou ca see is bellshaped..8 Magial pobabilit desit fuctio The magial p.d.f of is defied as f ( ) ad is the equatio of a cuve called the p.d.f. cuve of. P( a b) is a aea ude the p.d.f. cuve ad so P( a b) f ( ) b d (as i Chapte, sectio.8). a It ca be obtaied fom the joit p.d.f. b a sigle itegatio, as follows: f ( ) f d. The magial p.d.f of is defied as f ( ) ad is the equatio of a cuve called the p.d.f. cuve of. P( c d) is a aea ude the p.d.f. cuve ad so d. P( c d) f ( ) d c It ca be obtaied fom the joit p.d.f. b a sigle itegatio, as follows: ad f ( ) f d
5 .9 Coditioal pobabilit desit fuctios The coditioal p.d.f of give is deoted b f ( ) ad defied as f ( ) f ( ) 5 ( ) f wheeas the coditioal p.d.f of give is deoted b f ( ) ad defied as f ( ) f ( ) f ( ) f f.0 Joit pobabilit distibutio fuctio As i.5 the joit (cumulative) pobabilit distibutio fuctio (c.d.f.) is deoted b F(, ) ad is defied as F(, ) P( ad ) but F(, ) i the cotiuous case is the volume ude the p.d.f. suface fom to ad fom to, so that F( ) v u, v u f u v du dv The magial c.d.f. s ae defied as i.5 ad ca be obtaied fom the joit distibutio fuctio F(, ) as follows: F F ( ) F( MA ) ( ) F( ), whee MA is the lagest value of ad MA, whee MA is the lagest value of.. Impotat coectios betwee the p.d.f s ad the joit c.d.f. s. (i) The joif p.d.f. f (, ) F (ii) The magial p.d.f s ca be obtaied fom the magial c.d.f. s as follows: the magial p.d.f. of f ( ) the magial p.d.f. of f ( ) df d df d ( ) ( ) o F ( ), o F ( ). Ae ad idepedet? ad ae idepedet adom vaiables if eithe (a) F(, ) F ( ) F ( ) ; o
6 (b) f(, ) f ( ) f ( ) ; o 6 (c) f ( ) fuctio of ol o equivaletl f ( ) fuctio of ol Eample.5 The joit distibutio fuctio of ad is give b F + 0, 0 othewise (i) Fid the magial distibutio ad desit fuctios. (ii) Fid the joit desit fuctio. (iii) Ae ad idepedet adom vaiables? Eample.6 ad have the joit pobabilit desit fuctio 8 f, 7 (a) Deive the magial distibutio fuctio of. (b) Deive the coditioal desit fuctio of give (c) Ae ad idepedet? Give: Give: Joit desit f. f (,) Joit distibutio f. F(, ) Itegate w..t Diffeetiate (patiall) ad w..t. ad Joit distibutio f. F(, ) Joit desit f. f (, ) v v u u f u v du dv F.
7 Eample.6(b) ad (c) 7 Solutio Fom.9 the coditioal p.d.f of give is deoted b f ( ) ad defied as f ( ) f ( ) ( ) f ad ad f ( ) is the magial p.d.f. of. We kow f ( ) fid f ( ). Thee ae two was ou ca fid f ( ) f whee f (, ) is the joit p.d.f. of, 8 7 so we eed to. The fist wa ivolves itegatio ad the secod wa ivolves diffeetiatio. I will do both was to show ou how to use the diffeet esults we have hee but ou should alwas choose the wa ou fid easiest i.e ou would ot be epected to fid f ( ) both was i a assessed wok. Method Fom.8 f ( ) f (, ) d so f ( ) 8 7 d 8 7 d Method Fom. f ( ) df ( ) d Fom.0 F ( ) F( ) F F whee F ( ) MA, whee MA ( ) F ad fom pat (a), F( ) ( ) Hece f ( ) 4 4, ( ) d 4 d 4 is the magial c.d.f of. is the lagest value of, so 4, ( ) 8 hece F ( ) so 4. as with method. Now theefoe the coditioal desit fuctio of give, f ( ) is give b So ( ) f f ( ) (, ) f ( ) f 7 ad 0 othewise (c) Now f ( ) is a fuctio of ol, so usig esult.(c), ad ae idepedet. Notice also that f ( ) f ( ) which ou would epect if ad ae idepedet.
8 . Epectatios ad vaiaces 8 Discete adom vaiables ( ) p(, ) p ( ) E p ( ) p(, ) p ( ) E p Eamples,...,... Hece Va() E( ) ( E( )), Va() E( ) ( E( ) Cotiuous adom vaiables ( ) ( ) ( ) etc. E f, d d f d,... ( ) ( ) ( ) E f, d d f d,... Eamples.4 Epectatio of a fuctio of the.v.'s ad Cotiuous ad Discete ad E[ g] g f dd e. g. E f dd E[] f (, ) dd..5 Covaiace ad coelatio Covaiace of ad is defied as follows : Cov (,) σ E()  E()E(). Notes (a) If the adom vaiables icease togethe o decease togethe, the the covaiace will be positive, wheeas if oe adom vaiable iceases ad the othe vaiable deceases ad vicevesa, the the covaiace will be egative. (b) If ad ae idepedet.v's, the E() E()E() so cov(, ) 0. Howeve if cov(,) 0, it does ot follow that ad ae idepedet uless ad ae
9 9 Nomal.v's. Coelatio coefficiet ρ co(,) Cov (, ). σ σ Note (a) The coelatio coefficiet is a umbe betwee  ad i.e.  ρ (b) If the adom vaiables icease togethe o decease togethe, the ρ will be positive, wheeas if oe adom vaiable iceases ad the othe vaiable deceases ad vicevesa, the ρ will be egative. (c) It measues the degee of liea elatioship betwee the two adom vaiables ad, so if thee is a oliea elatioship betwee ad o ad ae idepedet adom vaiables, the ρ will be 0. ou will stud coelatio i moe detail i the Ecoometic pat of the couse with David Wite. Eample.7 I Eample. ae ad coelated? Solutio Below is the joit o bivaiate pobabilit distibutio of ad : The magial distibutios of ad ae Total p P( ) o ( ) ad P( ) o ( ) 0 0 Total p Eample. 8 I Eample.6 (i) Calculate E(), Va(), E() ad cov(,). (ii) Ae ad idepedet?.4 Useful esults o epectatios ad vaiaces (i) E( a + b) ae( ) + be( ) whee a ad b ae costats.
10 0 (ii) Va( a + b) a Va( ) + b Va( ) + ab cov. Result (i) ca be eteded to a adom vaiables,,..., E a + a a a E + a E a E ( ) ( ) ( ) ( ) Whe ad ae idepedet, the (iii) Va( a + b) a Va( ) + b Va( ) so cov(, ) 0 (iv) E( ) E( ) E( ) Results (iii) ad (iv) ca be eteded to a idepedet adom vaiables,,..., (iii)* Va( a a... a ) ( ) + ( ) ( ) a Va a Va a Va (iv)* E(... ) E( ). E( )... E( )
11 .5 Combiatios of idepedet Nomal adom vaiables Suppose ~ N ( µ, σ ), ~ N( µ, σ ), ~ N ( µ, σ ),...ad ~ N ( µ, σ ),,..., ae idepedet adom vaiables, the if a + a + a a whee a, a... a ae costats, ~ N( a µ + a µ + a µ a µ, a σ + a σ a σ ) i.e. ~ N( a µ, a σ ). i i i i
12 I paticula, suppose,... fom a adom sample fom a Nomal populatio with mea µ ad vaiace σ, µ µ µ... µ µ ad σ σ... σ σ. ~ N( a µ, a σ ). i i Futhe, suppose that a a a... a the ad ~ N( µ, σ ).
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